fuse blowing sounds like a short or pulling too much current on that circuit.
Have you found what that circuit serves?
Don't know what other suggestions you've received for the gennie but the not moving vs moving detail would seem yo indicate a loose connection or component in the gennie itself. Grab a PDF for your model & year generator. It should have a pretty well organized troubleshooting flowchart or steps.
I had a bout of intermittent cutoffs on my Generac a short after i bought my rig. After a couple weeks chasing all manner of red herrings, it turned out the spring inside the glass fuse holder had popped out one day when i checked the fuse at night....the fuse would move around enough to let it run dir a minute then die. I even replace a couple solinoids. Turned out to be a 3ยข spring. I bought a new fuse holder because i couldn't keep hunting for that spring...
good luck...it takes a minute but it WILL get better as ypu get to know the quirks-mode of your rig!